Orientation data for moduli spaces of coherent sheaves over Calabi–Yau 3-folds


Joyce, D
Upmeier, M

Publication Date: 

11 February 2021


Advances in Mathematics

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<p>Let X be a compact Calabi–Yau 3-fold, and write M,M
for the moduli stacks of objects in coh(X), Dbcoh(X). There
are natural line bundles KM → M, KM → M, analogues
of canonical bundles. Orientation data on M,M is an
isomorphism class of square root line bundles K1/2
M , K1/2
M ,
satisfying a compatibility condition on the stack of short exact
sequences. It was introduced by Kontsevich and Soibelman
[35, §5] in their theory of motivic Donaldson–Thomas invariants, and is also important in categorifying Donaldson–
Thomas theory using perverse sheaves.</p>
<p>We show that natural orientation data can be constructed for
all compact Calabi–Yau 3-folds X, and also for compactlysupported coherent sheaves and perfect complexes on noncompact Calabi–Yau 3-folds X that admit a spin smooth
projective compactification X → Y . This proves a longstanding conjecture in Donaldson–Thomas theory.</p>
<p>These are special cases of a more general result. Let X be
a spin smooth projective 3-fold. Using the spin structure we
construct line bundles KM → M, KM → M. We define
spin structures on M,M to be isomorphism classes of square
roots K1/2
M , K1/2
M . We prove that natural spin structures exist
on M,M. They are equivalent to orientation data when X is
a Calabi–Yau 3-fold with the trivial spin structure.</p>

<p>We prove this using our previous paper [33], which constructs
‘spin structures’ (square roots of a certain complex line bundle
P → BP ) on differential-geometric moduli stacks BP of connections on a principal U(m)-bundle P → X over a
compact spin 6-manifold X.</p>

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Publication Type: 

Journal Article